High Resolution DOA Estimation Method in MIMO Radar

Abstract

The resolution of existing DOA estimation methods in multiple-input and multiple-output (MIMO) radar is restricted by the number of antennas. In this paper, a new method to expand the number of valid receiving antennas virtually in MIMO radar without adding any extra actual antenna is proposed. In comparison with the existing DOA estimation methods, e.g. Capon method and Compressive Sensing method, the proposed method show its superiority if the autocorrelation property of transmitting waveform keeps good. In order to relax such limitation, we further proposed two improved version of the proposed method. Simulation results validate the superiority of the proposed methods.

Appendix

For simplifying (12) and (13), we set,

$$ x_{1} (t) = x(t - 2d_{1} (0)/c) $$

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$$ x_{2} (t) = x(t - 2d_{2} (0)/c) $$

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$$ \gamma_{11} = \beta_{1} \cdot \;\exp ( - j2\pi f(d_{11}^{t} (t) + d_{11}^{r} (t))/c) $$

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$$ \gamma_{12} = \beta_{2} \cdot \;\exp ( - j2\pi f(d_{12}^{t} (t) + d_{12}^{r} (t))/c) $$

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$$ \gamma_{21} = \beta_{1} \cdot \;\exp ( - j2\pi f(d_{11}^{t} (t) + d_{21}^{r} (t))/c) $$

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$$ \gamma_{22} = \beta_{2} \cdot \;\exp ( - j2\pi f(d_{12}^{t} (t) + d_{22}^{r} (t))/c) $$

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We assume that the reflection coefficients of all targets are equal, i.e., $\beta_{1} = \beta_{2} = 1$. Then Eqs. (12) and (13) can be rewritten as, respectively,

$$ y_{1m} (t) = \gamma_{11} x_{1} (t) + \gamma_{12} x_{2} (t) + n_{1} (t) $$

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$$ y_{2m} (t) = \gamma_{21} x_{1} (t) + \gamma_{22} x_{2} (t) + n_{2} (t) $$

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The auto-correlation property of transmitting waveforms is denoted as,

$$ \int_{t = 0}^{T} {x(t - \tau_{1} )} x^{*} (t - \tau_{2} )dt/T \approx \left\{ {\begin{array}{*{20}c} {\varepsilon ,} & {\tau_{1} \ne \tau_{2} } \\ {1,} & {\tau_{1} = \tau_{2} } \\ \end{array} } \right. $$

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If the auto-correlation property keeps good, i.e., $\varepsilon$ is very small, $\gamma_{ij}$ can be recovered as,

$$ \begin{aligned} \hat{\gamma }_{11} & = \frac{1}{T}\int_{0}^{T} {x_{{_{1} }}^{*} (t)y_{1m} (t)dt} \\ & = \frac{{\gamma_{11} }}{T}\int_{0}^{T} {x_{{_{1} }}^{*} (t) \cdot x_{1} (t)} dt + \frac{{\gamma_{12} }}{T}\int_{0}^{T} {x_{{_{1} }}^{*} (t) \cdot x_{2} (t)} dt + \frac{1}{T}\int_{0}^{T} {x_{{_{1} }}^{*} (t) \cdot n_{1} (t)} dt \\ & = \gamma_{11} + \tilde{\gamma }_{11} \\ \end{aligned} $$

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$$ \begin{aligned} \hat{\gamma }_{12} & = \frac{1}{T}\int_{0}^{T} {x_{{_{2} }}^{*} (t) \cdot y_{1m} (t)dt} \\ & = \frac{{\gamma_{11} }}{T}\int_{0}^{T} {x_{{_{2} }}^{*} (t) \cdot x_{1} (t)} dt + \frac{{\gamma_{12} }}{T}\int_{0}^{T} {x_{{_{2} }}^{*} (t) \cdot x_{2} (t)} dt + \frac{1}{T}\int_{0}^{T} {x_{{_{2} }}^{*} (t) \cdot n_{1} (t)} dt \\ & = \gamma_{12} + \tilde{\gamma }_{12} \\ \end{aligned} $$

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$$ \begin{aligned} \hat{\gamma }_{21} & = \frac{1}{T}\int_{0}^{T} {x_{{_{1} }}^{*} (t) \cdot y_{2m} (t)} dt \\ & = \frac{{\gamma_{21} }}{T}\int_{0}^{T} {x_{{_{1} }}^{*} (t) \cdot x_{1} (t)} dt + \frac{{\gamma_{22} }}{T}\int_{0}^{T} {x_{{_{1} }}^{*} (t) \cdot x_{2} (t)} dt + \frac{1}{T}\int_{0}^{T} {x_{{_{1} }}^{*} (t)n_{2} (t)} dt \\ & = \gamma_{21} + \tilde{\gamma }_{21} \\ \end{aligned} $$

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$$ \begin{aligned} \hat{\gamma }_{22} & = \frac{1}{T}\int_{0}^{T} {x_{{_{2} }}^{*} (t) \cdot y_{2m} (t)} dt \\ & = \frac{{\gamma_{21} }}{T}\int_{0}^{T} {x_{1} (t) \cdot x_{{_{2} }}^{*} (t)} dt + \frac{{\gamma_{22} }}{T}\int_{0}^{T} {x_{{_{2} }}^{*} (t) \cdot x_{2} (t)} dt + \frac{1}{T}\int_{0}^{T} {x_{{_{2} }}^{*} (t)n_{2} (t)dt} \\ & = \gamma_{22} + \tilde{\gamma }_{22} \\ \end{aligned} $$

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where $\tilde{\gamma }_{11} = \varepsilon \gamma_{12} + \varepsilon^{\prime}\sigma_{1}$, $\tilde{\gamma }_{12} = \varepsilon \gamma_{11} + \varepsilon^{\prime}\sigma_{1}$, $\tilde{\gamma }_{21} = \varepsilon \gamma_{21} + \varepsilon^{\prime}\sigma_{2}$ and $\tilde{\gamma }_{22} = \varepsilon \gamma_{21} + \varepsilon^{\prime}\sigma_{2}$ are estimation errors. $\varepsilon^{\prime}$ is the value of the cross-correlation between the transmitting waveform and noise signal.